Let S be a set of positive integers and define a relation ρ on S as follows: ∀a, b ∈ S, aρb if and only if a ≤ b and a and b have the same number of positive divisors.
(a) Let S = {1,2,3,4,5,6,7,8}. Draw an arrow diagram of ρ on the set S.
(b) Let S = Z+. Prove that (S, ρ) is a partial order.
I am stuck on part a) i am not to sure on how to draw an arrow diagram of ρ while for part b), May i know how do i prove a statement to be a partial order.
a) For this part you should draw an arrow from $i$ to $j$ if and only if $a \rho b$. As an example, you draw an arrow from $3$ to $5$ as $3 \leq 5$ and they have the same number of positive divisors.
b) You need to prove the axioms of partial order: i) reflexivity $a \rho a$. This is clear as $a \leq a$ and they have the same number of positive divisors.
ii) antisymmetry If $a \rho b$ and $ b \rho a$, then $a=b$. This follows easy $ a \rho b$ and $b \rho a$ implies $a \leq b$ and $b \leq a$ respectively, so $a=b$.
iii) transitivity If $a \rho b$ and $ b \rho c$, then $a \rho c$. I will this as an exercise.